Quadratic forms with p-adic coefficients are mathematical functions that involve a combination of squares of variables and coefficients that are integers in the p-adic number system.
P-adic quadratic forms are different from traditional quadratic forms because they are defined over the p-adic number system instead of the real or complex number system. This means that the coefficients and variables in p-adic quadratic forms take on values in the p-adic numbers, which have unique properties and behaviors compared to traditional numbers.
Quadratic forms with p-adic coefficients have a wide range of applications in number theory, algebraic geometry, and cryptography. They are also used in the study of quadratic forms over non-archimedean fields, which has implications in areas such as algebraic number theory and representation theory.
P-adic quadratic forms are important in mathematics because they provide a powerful tool for studying number systems that are not well understood using traditional methods. They also have connections to various areas of mathematics, making them a useful tool for interdisciplinary research.
Some open problems and current research in the field of p-adic quadratic forms include generalizing the theory to higher dimensions, studying the behavior of quadratic forms over more general p-adic fields, and investigating connections to other areas of mathematics such as modular forms and automorphic representations.